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Theorem reseq1d 4786
Description: Equality deduction for restrictions. (Contributed by NM, 21-Oct-2014.)
Hypothesis
Ref Expression
reseqd.1  |-  ( ph  ->  A  =  B )
Assertion
Ref Expression
reseq1d  |-  ( ph  ->  ( A  |`  C )  =  ( B  |`  C ) )

Proof of Theorem reseq1d
StepHypRef Expression
1 reseqd.1 . 2  |-  ( ph  ->  A  =  B )
2 reseq1 4781 . 2  |-  ( A  =  B  ->  ( A  |`  C )  =  ( B  |`  C ) )
31, 2syl 14 1  |-  ( ph  ->  ( A  |`  C )  =  ( B  |`  C ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1314    |` cres 4509
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 681  ax-5 1406  ax-7 1407  ax-gen 1408  ax-ie1 1452  ax-ie2 1453  ax-8 1465  ax-10 1466  ax-11 1467  ax-i12 1468  ax-bndl 1469  ax-4 1470  ax-17 1489  ax-i9 1493  ax-ial 1497  ax-i5r 1498  ax-ext 2097
This theorem depends on definitions:  df-bi 116  df-tru 1317  df-nf 1420  df-sb 1719  df-clab 2102  df-cleq 2108  df-clel 2111  df-nfc 2245  df-v 2660  df-in 3045  df-res 4519
This theorem is referenced by:  reseq12d  4788  fun2ssres  5134  funcnvres2  5166  funimaexg  5175  fresin  5269  offres  5999  tfrlemisucaccv  6188  tfrlemi1  6195  tfr1onlemsucaccv  6204  tfrcllemsucaccv  6217  freceq1  6255  freceq2  6256  fseq1p1m1  9825  setsresg  11903  setscom  11905  dvcoapbr  12746
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